Higher-dimensional geometric sigma models

PHYSICAL REVIEW D, VOLUME 60, 025003

Higher-dimensional geometric sigma models

M. Vasilic*Department of Theoretical Physics, Institute Vincˇa, P.O. Box 522, 11001 Beograd, Yugoslavia

~Received 20 November 1998; published 14 June 1999!

Geometrics models have been defined as geometric theories of metric excitations of a given backgroundgeometry, and then covariantized by identifying the coordinates of space-time with a set of scalar fields. Byconstruction, these theories have the property of accommodating both the scalar matter of pure geometricorigin and a ground state specified in advance. Using this fact, one can build a Kaluza-Klein geometricsmodel by specifying the background metric of the formM43Bd, thus obtaining a theory free of the classicalcosmological constant problem. Previously exploited ideas to use scalar fields in the form of a nonlinearsmodel coupled to gravity to trigger the compactification failed to give massless gauge fields after dimensionalreduction. In this paper, a modified geometrics model is suggested, which reconciles the masslessness of thegauge fields with the zero value of the cosmological constant.@S0556-2821~99!04310-6#

PACS number~s!: 11.10.Kk, 04.50.1h

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I. INTRODUCTION

The classical cosmological constant problem of KaluKlein theories@1# whose internal manifold is not Ricci flat ia long standing one. In the conventional Kaluza-Klein trement, the internal manifold is chosen in such a way thatisometries define the internal symmetries of the theory.the same time, a ground state in the form of the direct pruct of the 4-dimensional flat space-time with a compact, nflat internal space does not satisfy the classical EinstHilbert equations of motion. The attempt do solve probleby adding a cosmological term has failed. Indeed, to repduce some known gauge couplings, the cosmological cstant is constrained to be of the order of the Planck msquared, which strongly disagrees with the observedverse.

Among a variety of existing approaches to this problewe shall focus our attention on those which use matter fieto trigger spontaneous compactification. At the same tiwe do not want to lose the geometric character of our theo.How can we reconcile these two requirements? Noticethis respect, that the so called geometrics models have beendefined in@2# as purely geometric theories of scalar fielcoupled to gravity. By construction, these scalar fields ornate from the coordinates of space-time, and, as a coquence, can be gauged away. We are left then withspace-time metric as the only variable in the theory, andnoncovariant field equations which govern its dynamics. Tactual procedure of constructing geometrics models takesthe opposite direction. It begins by specifying a fixed met

field configurationg° mn as the possible vacuum of the gemetric theory to be built. Then, the dynamics is chosen fra variety of different possibilities. The simplest one is

postulate Einstein-like equations of the formRmn5R° mn ,

where the fixed functionR° mn stands for the Ricci tensor o

the background metricg° mn . These are the noncovariant fie

*E-mail address: [email protected]

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equations whose nonvanishing right-hand side actuallyde-finesmatter, and which, by construction, possess the class

solution gmn5g° mn . Whether this solution can representphysically acceptable vacuum of the theory is nota prioriclear. Its classical stability, for example, strongly dependsthe choice of the background metric. The asymptotic stabiof localized, static, spherically symmetric excitations of a fl4-geometry has already been analyzed in@2#. In general,however, it should be separately checked which particubackground geometry can be stabilized by our simple n

covariant dynamicsRmn5R° mn . The noncovariant characteof this theory stems from the fact that by fixing the form

R° mn one also fixes the very coordinate system. It is brouto a covariant form by employing a new set of coordinatsayf i5f i(x), to fix the Ricci tensor on the right-hand sidThen, the equations of motion take the form of a nonlineasmodel coupled to gravity, with the scalar sector consistingas many scalar fieldsf i(x) as the number of space-timdimensions. By choosing the gaugef i5xi , the field equa-tions are brought back to the noncovariant formRmn

5R° mn .As a result of the above procedure, an action functio

describing gravity in interaction with a nonlinears model isassociated with each given background metric. Usingfact, one can build a Kaluza-Klein geometrics model byspecifying the background geometry in the form of the dirproduct of a 4-dimensional Minkowski spaceM4 with theinternal d-dimensional spaceBd. The resulting theory willnecessarily possess the classical solutionM43Bd, and,therefore, be free of the classical cosmological constproblem. An action functional of this kind has already bediscussed in literature. The authors of Refs.@3# and@4# haveemployed scalar fields in the form of a nonlinears model totrigger the compactification. It is not difficult to see that themodel is a particular example of a geometrics model. Itturns out, however, that, although solved the classical cmological constant problem, it failed to give massless gafields. We shall modify the model of@3# and@4# in the spiritof @2#, and reconcile the masslessness of the gauge fi

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M. VASILIC PHYSICAL REVIEW D 60 025003

with the zero value of cosmological constant.The layout of the paper is as follows. In Sec. II, we sh

analzse the model suggested in@3# and@4# from the point ofview of geometrics models. We shall readily use the gaufreedom, and fix all the scalar fields in the theory therereducing the field equations to the purely geometric, non

variant form RMN5R° MN , the functionR° MN on the right-hand side standing for the Ricci tensor of the Kaluza-Klbackground metricM43Bd. The linearized version of thetheory of small excitations of this background metric turout to be easily averaged over the internal coordinates.resulting effective 4-dimensional equations of motion athen shown to necessarily contain massive terms in the seof gauge fields. Although the result is not new, the analyof this model along the lines of Ref.@2# has proved to bevery useful in detecting the cause of the appearance of msive Yang-Mills fields. A careful examination of the ave

aged noncovariant field equationsRMN5R° MN clearly sug-gests a simple modification of this model. In Sec. III, wshall see that the Yang-Mills sector of the effecti4-dimensional equations of motion obtained by averaging

equationsRMN5R° MN contains no massive gauge fields. Tfact that the new model uses Ricci tensors with upper indturns out to be crucial. However, it happens to be difficultconstruct the corresponding Lagrangian. This is why we sgest another modification of the model in Sec. IV. By add

terms proportional to (GMN2G° MN) to the equations of motion, we shall certainly not lose the good property of omodel to have vanishing cosmological constant. Indeed, s

equations possess the classical solutionGMN5G° MN , and we

choose the metricG° MN to be of the needed typeM43Bd.We shall be able to demonstrate that, after the dimensioreduction, our higher dimensional Lagrangian gives the sdard Einstein, Yang-Mills and Klein-Gordon sectors. Tscalar excitations of the internal manifold turn out to bemassive, with masses of the order of the Planck mass.analysis is confined within the linear approximation of ttheory of small excitations of the background metric.

In Sec. V, we shall covariantize our model. Following tideas of geometrics models of Ref.@2#, we shall employ aset of 41d scalar fieldsVA(X) representing the new coordinates of the space-time. When the fixed quantities of

model, such asR° MN andR° , are computed in this new coordinate system, the theory becomes generally covariant.obtained action functional has the form of a nonlinearsmodel coupled to gravity, but is not of a standard type. Tkinetic term of the corresponding Lagrangian turns out toa nonpolynomial function of the scalar field derivatives. Itthe consequence of the presence of background quanwith upper indices, whose transformation laws employinverse ofV ,M

A rather thanV ,MA itself. In any case, the non

polynomial character of the theory is not essential owingthe possibility that the scalar fields are gauged away.shall still be able to bring the field derivatives to a polynmial form by introducing a set of auxiliary fields. The obtained theory is briefly compared to the conventional non

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ear s models. It is shown how a degenerate target spmetric leads to an extra symmetry of the action, but doesallow for the reduction of the number of the scalar fields.the effective 4-dimensional theory, however, this symmewill admit an additional gauge fixing. The analysis of the fuinteracting covariant theory is left for future investigations

Section VI is devoted to concluding remarks.

II. COORDINATES AS MATTER FIELDS

The model the authors of Refs.@3# and @4# discuss con-sists of Einstein gravity in 41d dimensions coupled to anonlinears model:

I 52k2E d41dXA2G@R2Fi j ~V!V ,Mi V ,N

j GMN#.

~2.1!

The scalar fieldsV i , i 51,2, . . . ,d, are thought of as coordinates of ad-dimensional compact Riemannian manifoldBd

with Ricci tensor Fi j (V), while the coordinatesXM

[(xm,ym) parametrize a (41d)-dimensional space-timewith metric GMN . The indices run as follows:

M ,N50, . . . ,31d, m,n50, . . . ,3, m,n51, . . . ,d.

Notice that the number of scalar fields equals the numbecompact dimensions of the space-time. The equations oftion for this theory possess a classical solution of the for

GMN5G° MN , Vm5ym, ~2.2!

where

G° MN[S hmn 0

0 fmn~y!D , ~2.3!

andfmn(y) stands for the metric ofBd. The scalar sector othe solution~2.2! is obviously topologically nontrivial sinceit is described by a degree one mapping fromBd to Bd. Atthe same time, the metric~2.3! has the form of the direcproduct of the 4-dimensional Minkowski space-time withcompact internal space, as desired. If we restrict our attento the physics of small excitations of this classical solutiowe can always choose the space-time coordinates to fixVm

5ym. Then, the action functional~2.1! reduces to

I 52k2E d41dXA2G~RMN2R° MN!GMN, ~2.4!

where R° MN stands for the Ricci tensor of the backgroumetric ~2.3!. This is a purely geometric but noncovariatheory whose physical content is fully contained in the eqtons of motion

RMN5R° MN . ~2.5!

Comparing it with the results of@2#, we can see that theabove theory is a particular example of a geometrics modelbased on the background metric~2.3!. The covariantization

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HIGHER-DIMENSIONAL GEOMETRIC SIGMA MODELS PHYSICAL REVIEW D60 025003

of the equations~2.5! in the spirit of @2# introduces 41dscalar fields in the form of a nonlinears model. Indeed,using the new coordinatesVA5VA(X), A50, . . . ,31d, tofix the background value of the Ricci tensor, the equatio~2.5! turn into

RMN5FAB~V!V ,MA V ,N

B ~2.6!

with FAB(V)[R° AB(V), so that, after the new coordinateare identified with the old ones,VA5XA, the equations~2.6!reduce to Eq.~2.5! again. It is owing to the special form o

G° MN as given by Eq.~2.3! that only d out of 41d scalarfields survive. The equations~2.6! are easily seen to followfrom the action functional~2.1!. In addition, the matter fieldequations obtained by varying Eq.~2.1! with respect toV i

are not independent ones, but rather follow from Eq.~2.6!and the Bianchi identities (RMN2 1

2 GMNR) ;N50. Conse-quently, the purely geometric, noncovariant equations~2.5!carry the full physical content of the theory given by E~2.1!.

As mentioned in the Introduction, the model~2.1! solvesthe classical cosmological constant problem but fails to gmassless Yang-Mills fields after the dimensional reductiWhen rewritten as Eq.~2.5!, the theory automatically take

care of the cosmological term. Indeed, the metricG° MN is bydefinition a solution to the equations of motion~2.5!. Toanalyze the spectar of the corresponding effect4-dimensional theory, we shall use the standard 41d decom-position @5# of the metricGMN :

GMN[S gmn1Bmk Bn

l ukl Bmk ukn

Bnkukm umn

D . ~2.7!

For the perturbations of the background metricG° MN weadopt the notation

gmn5hmn1hmn , umn5fmn1wmn . ~2.8!

Substituting these expressions into Eq.~2.5! we find

Rnm50, ~2.9a!

Rnm52R° nlB

lm1O~2!, ~2.9b!

Rnm5R° n

m2R° nlwml1O~2!, ~2.9c!

where R° mn is the d-dimensional Ricci tensor of the background metricfmn , andO(2) stands for the terms quadratin field variables. For our purposes, the mixed componeRN

M of the Ricci tensor turn out to be more convenient.deed, it is the mixed components which, after the decomsition ~2.7! is employed, give the standard Einstein aYang-Mills terms~Appendix!. The expressions on the righhand side of Eq.~2.9! then measure the deviation of outheory from the standard case. In particular, we shall see

the termR° nlBlm is responsible for the appearance of mass

gauge fields.

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To obtain effective 4-dimensional equations of motiowe shall average Eqs.~2.9a!–~2.9c! over the internal coordi-nates. The average of ad-scalar S, as defined by

^S&[E ddyA2uS

E ddyA2u

5

E ddyA2fS

E ddyA2f

1O~2! ~2.10!

is also ad-scalar. However, a simple definition of the kinfor d-vectors andd-tensors does not exist. This is why whave to project Eqs.~2.9a!–~2.9c! on an appropriately chosebasis inBd before we use Eq.~2.10! to average them.~Thiskind of dimensional reduction has already been used in@6# toobtain effective 4-dimensional equations of motion outdimensionally continued Euler forms.! As is customary, letus suppose thatBd is a homogeneous space withm KillingvectorsKa

l (y), a51, . . . ,m, which form a~generally over-complete! basis inBd. Using the decomposition

Bnm5Ka

mAna , wmn5KamKbnw

ab, ~2.11!

and projecting the vector and tensor equations~2.9a!–~2.9c!on the Killing basis, we obtain an equivalent set ofd-scalarfield equations~A2a!–~A2c!. Although basically linear, theaveraged field equations will contain terms of the ty^S1S2& owing to the presence ofy-dependent coefficientsThe averageS1S2& cannot generally be expressed in termof ^S1& and ^S2&. However, our internal manifold is of thePlanckian size, and it is not unreasonable to restrictanalysis to solutions which slowly vary iny directon. In thatcase, the product of averages^S1&^S2& becomes the leadingterm in the decomposition

^S1S2&5^S1&^S2&1D12

so thatD12 can be regarded as a small correction. Using tfact and the fact that averages of covariantd-divergencesvanish, we obtain the following effective 4-dimensionequations of motion:

Rmn11

2w ,mn5B, ~2.12a!

gab]nFbmn12mabAbm5B, ~2.12b!

sabcdhwcd1mabcdwcd5B. ~2.12c!

Here,Rmn is the Ricci tensor of the 4-metricgmn , h is thecorresponding d’Alembertian, andFmn

a [Am,na 2An,m

a 1O(2)is the gauge field strength for the gauge fieldsAm

a . The barover a quantity denotes its expectation value as definedEq. ~2.10!, and B[D1O(2). The coefficients in Eqs.~2.12a!–~2.12c! are expectation values of products of thKilling vectors and their covariant derivatives, as is explitly shown in the Appendix. In particular,

gab[^KamKbm&, mab[^Ka

mKbnR° mn&.

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We see that the fieldsAma are generally massive with mass

proportional to the curvature of the internal manifoldBd. Soare the scalar excitationswab, with the exception of

w[gabwab5^wm

m&1D

which turns out to be massless. We can use that fact tocale the metric gmn according to gmn[(11w/2)gmn ,whereby Eq.~2.12a! takes the standard Einstein formRmn

5B.

III. MASSLESS GAUGE FIELDS

By inspecting the field equations~2.9a!–~2.9c!, especiallyafter the decomposition~2.7! is employed, we notice that thmixed componentsRn

m of the Ricci tensor are those whichupon averaging, give the expected Yang-Mills terms. Tnonvanishing right-hand side of Eq.~2.9b! then measures thdeviation of our theory from the standard case. In particu

the termR° nlBlm makes the gauge fields massive. The si

plest way to get rid of it is to postulate the equations

motion of the formRNM5R° N

M . These, however, are not symmetric, and can only be used within the vielbein formalisFor this reason, we shall concentrate our attention onsymmetric field equations of the form

RMN5R° MN. ~3.1!

Notice that the field equations~3.1! are not equivalentto Eq.~2.5! as it might look at first sight. The background valuethe Ricci tensor on the right-hand side of Eq.~3.1! is a fixednumerical function determined by the form of the bacground metric. Its indices are lowered by the backgrou

metric G° MN , while the indices of the variable Ricci tensoon the left-hand side are lowered by the variable me

GMN . Consequently, RMN[GMLGNKRLK5GMLGNKR° LK

ÞG° MLG° NKR° LK[R° MN . Therefore, the physics of Eq.~3.1!may differ from that of Eq.~2.5!.

It is not difficult to show that the corresponding theocontains no massive gauge fields. Indeed, by rewritingequations~3.1! in terms of the mixed components of thRicci tensor, we find

Rnm50, ~3.2a!

Rnm50, ~3.2b!

Rnm5R° n

m1R° mlwnl . ~3.2c!

Owing to the direct product structure of the background mric ~2.3!, the critical term on the right-hand side of Eq.~3.2b!disappears. As a consequence, the averaged equationRn

m

50 will accommodate the standard massless Yang-Mfields. The effective 4-dimensional equations of motionobtained by the exact procedure described in Sec. II. Onereadily use the results of the Appendix, and just add

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minor corrections as seen by comparing the field equati~2.9a!–~2.9c! with those of Eq.~3.2!. This leads to

Rmn11

2w ,mn5B, ~3.3a!

]nFamn5B, ~3.3b!

sabcdhwcd1mabcd8 wcd5B, ~3.3c!

where the group metricgab is used to raise and lower thgroup indices. We see that there are no massive terms inYang-Mills sector of Eqs.~3.3a!–~3.3c!. The mass matrixmabcd8 now differs from that of Eq.~2.12c! by the presence oan additional term, as in Eq.~A4!. In particular, the formerlymassless scalar fieldw acquires mass of the order of thPlanck mass. If we choose ourBd to be an Einstein manifold

sayR° mn5lfmn , we shall find (h24l)w5B. As a conse-quence ofhwÞ0, the local rescalings of the metricgmn

cannot bring Eq.~3.3a! into the standard Einstein form. Stilit is possible to fix the gauge]ncmn5 1

2 w ,m in the linearizedtheory (cmn[hmn2 1

2 hmnhll) thereby reducingRmn1 1

2 w ,mn

50 to hhmn50, as is customary. In this respect, notice thalthough the equations~3.1! are basically noncovariant, thestill possess a partial gauge symmetry as a consequencour special choice ofBd. Indeed, it is not difficult to checkthat the coordinate transformations

xm85xm8~x!, ym85ym1ea~x!Kam~y!

do not change the form of the equations of motion~3.1!. Themasslessness of the Yang-Mills sector of our field equatiis a natural consequence of the gauge symmetry inducethe above coordinate transformations. This is exactly wone could expect knowing how the same mechanism wowith conventional Kaluza-Klein theories. Strangely enougthe equations~2.5! do not possess this kind of symmetry, anit is not surprising that they accommodate massive gafields.

Before we covariantize the noncovariant field equatio~3.1!, we would like to define the corresponding action funtional. It turns out, however, that no obvious generalizatof Eq. ~2.4! exists. In the next section, we shall suggesLagrangian whose equations of motion differ from Eq.~3.1!but retain all their good features.

IV. LAGRANGIAN

The geometrics-model approach to the classical cosmlogical constant problem does not uniquely single outequations of motion in the form of Eq.~2.5! or Eq.~3.1!. One

can always add terms proportional to (GMN2G° MN) without

losing the classical solutionGMN5G° MN . We shall use thisfreedom to modify the equations~3.1! in a way which willallow for a simple construction of the corresponding Lgrangian. At the same time, we have to carefully choose

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correction in order not to lose the needed masslessness oYang-Mills sector.

A simple analysis along these lines takes us to the folloing noncovariant action functional:

I 52k2E d41dXA2G@R2R° 1R° MN~GMN2G° MN!#.

~4.1!

Varying it with respect toGMN gives the equations of motioof the desired form:

RMN5R° MN22

21dGMNR° LR~GLR2G° LR!. ~4.2!

As we can see, the correction to Eq.~3.1! is proportional to

(GMN2G° MN), leading to the theory free of the classical comological constant problem. At the same time, when rewten in terms of the mixed components of the Ricci tensor,equations~4.2! contain no massive terms in the Yang-Milsector. Indeed, using the 41d decomposition of the fieldequations, we find

Rnm52

2

21ddn

mR° i j w i j , ~4.3a!

Rnm50, ~4.3b!

Rnm5R° n

m1R° mlwnl22

21ddn

mR° i j w i j .

~4.3c!

As in Eqs.~3.2a!–~3.2c!, the crucial term on the right-hanside of the Eq.~4.3b! is missing. Notice also that the noncovariant field equations~4.2! possess the same kind of partigauge symmetry as Eq.~3.1! of the preceding section. Thuswe expect our Yang-Mills fields to be massless.

The effective 4-dimensional equations of motion are otained using the averaging procedure of Sec. II. To simpthe analysis, we shall choose our internal space in the fof an Einstein manifold

R° mn5lfmn

with l,0 in accordance with the adopted conventio@RNLR

M 5GNL,RM 2•••, diag(GMN)5(2,1, . . . ,1)]. Then,

making use of the results of the Appendix, the averagequations become

Rmn11

2w ,mn1

2l

d12hmnw5B, ~4.4a!

]nFamn5B,

~4.4b!

sabcdhwcd1mabcd9 wcd5B. ~4.4c!

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The gauge fieldsAma are obviously massless, but the sca

excitationswab have masses of the order of the Planck maIn particular, the scalar fieldw, appearing in Eq.~4.4a!, sat-isfies

S h28l

d12D w5B. ~4.5!

We see that the conventional choicel,0 ensures the correcsign for the mass term in Eq.~4.5!. Moreover, as opposed tthe case of Sec. III, Eq.~4.5! makes it possible to rescale thmetric gmn according to

gmn[S 11w

2D gmn1O~2!,

thereby bringing Eq.~4.4a! into the standard Einstein form

Rmn5B.

Now, we are in a position to say something more aboutclassical stability of our background geometryM43Bd. Theeffective 4-dimensional field equations~4.4a!–~4.4c! turnedout to have the form of the standard Einstein, Yang-Miand Klein-Gordon equations. The first two are known tolinearly stable against small fluctuations of the vacuumgmn

5hmn , Ama 50. As for the stability of the scalar excitation

wab of the Klein-Gordon sector of Eq.~4.4!, it depends onthe structure of the mass matrixmabcd9 . We have already

seen that the mass of the trace modew, as described by Eq~4.5!, is positive. However, the traceless part ofwab has to beanalyzed separately by calculatingmabcd9 for every particularchoice of the internal spaceBd.

The massesmabcd9 , as well as the coefficientssabcd andgab , are defined in the Appendix, as expectation valuesproducts of the Killing vectors and their covariant derivtives. They are constant tensors of the isometry group ofinternal manifoldBd. In the case ofBd5S2, for example,one finds

gab52

3dab ,

sabcd52

15dabdcd1

7

15~dacdbd1dbcdad!,

mabcd9 516

15dabdcd2

4

15~dacdbd1dbcdad!.

For simplicity, it has been setl521. The SO(3) tensorsabcd has the inverse defined through (s21)abcdscde f

[d (ea d f )

b . As a consequence, all the scalar fieldswab surviveas independent degrees of freedom in this theory. This isimprovement as compared to@6# where the classical cosmological constant problem has been solved at the expenslosing the kinetic terms of some scalar excitations. The mmatrix mabcd9 , being a constantSO(3) tensor itself, has the

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structure of the same type assabcd. The trace modew iseasily seen to have the positive mass equaling 2, in acdance with Eq.~4.5!. Unfortunately, all five traceless modeturn out to have the same negative mass2 4

7 . Thus, M4

3S2 is a classically unstable solution of the theory~4.1!. Thestability of a generic Kaluza-Klein background metric shoube checked separately for every particular choice ofBd. It isa difficult task by itself, and will not be addressed here.

V. COVARIANTIZATION

To covariantize the theory given by the action function~4.1!, we shall follow the ideas of reference@2#. As explainedin Sec. II, we shall use a new set of coordinates,VA

5VA(X), A50,1, . . . ,31d, to fix the background quantities of our model. Then, the covariantization is achievthrough the substitution

R° MN~X!→R° AB~V!]XM

]VA

]XN

]VB , R° ~X!→R° ~V!

in the equations of motion~4.2! or, equivalently, the La-grangian~4.1!. This gives

I 52k2E d41dXA2G

3FR1FAB~V!]XM

]VA

]XN

]VB GMN2V~V!G , ~5.1!

where the target metricFAB(V) and the potentialV(V) aredefined as

FAB~V![R° AB~V!, V~V![2R° ~V!.

The obtained action functional is generally covariant, andcorresponding equations of motion possess the classicalution

GMN5G° MN , VA5XA

with G° MN given by Eq.~2.3!. When restricted to the physicof small excitations of this classical solution, one can chothe coordinate system to enforce the gauge conditionVA

5XA, thereby taking us back to the noncovariant theory.see that the generally covariant theory~5.1! is equivalent toEq. ~4.1!, and, consequently, has all the good features ofpreceding section. In particular, the linearized equationsmotion, being of the form~4.4a!–~4.4c!, contain standardEinstein, Yang-Mills and Klein-Gordon sectors. The ondifference, as compared to the conventional Kaluza-Kltheories, is the absence of the massless scalar excitatiothe internal manifold.

The higher dimensional Lagrangian~5.1! looks like asmodel coupled to gravity, but is certainly not of a standatype. The derivatives of the scalar fieldsVA appear non-polynomially in the action. We can put them into a polynmial form by introducing a set of auxiliary fields. In particu

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d

eo-

e

e

ef

nof

d

lar, we need 41d vector fieldsbMA subject to the equations o

motion bMA 5V ,M

A . The easiest way to achieve this is to potulate the action functional:

I 52k2E d41dXA2G@R1FAB~V!hAMhMB

1lAM~bM

A 2V ,MA !2V~V!# ~5.2!

with hAM the inverse ofbM

A , and Lagrange multiplierslAM .

That it is indeed equivalent to Eq.~4.1! is shown by inspect-ing the equations of motion. One finds

bMA 5V ,M

A , ~5.3a!

lAM52FBC~V!hB

LhCMhLA , ~5.3b!

RMN5FAB~V!hAMhB

N22

d12GMNFFAB~V!hA

LhLB

21

2V~V!G , ~5.3c!

]V

]VA5]FBC

]VA hBMhMC2

1

2GMNG,L

MNlAL1lA,M

M .

~5.3d!

The auxiliary fieldsbMA andlA

M are fully expressed in termof GMN andVA, and carry no degrees of freedom. Equati~5.3d!, obtained by varying the action~5.2! with respect toVA, is not an independent equation of motion. It is easshown to follow from the Bianchi identities (RN

M

2 12 dN

MR) ;M[0 and Eqs.~5.3a! through ~5.3c!. It turns outthen that the content of the theory is fully contained in E~5.3c! with hA

M5]XM/]VA. After the space-time coordinateare chosen to fixVA5XA, the equations of motion boil downto Eq. ~4.2!, as expected. The fact that our theory has a pgeometric nature makes its nonpolynomial character bcally unessential.

The theory given by Eq.~5.2! or, equivalently, Eq.~5.1!differs in many aspects from the geometrics models of Ref.@2#, and, in that respect, from the model of Refs.@3# and@4#.First, the equations of motion~5.3! do not admit the topo-logically trivial solutionVA50 representing a nongeometrsector of the theory. Second, although our target meFAB(V) has vanishingFaB(a50, . . . ,3) components, westill need all 41d fields VA. In ordinary geometrics mod-

els, the rank of the Ricci tensorR° MN determines the numbeof necessary scalar fields. This is why we needed onlyd outof 41d scalar fields in the model defined by Eq.~2.1!. Here,we have to retain all the componentsVA, in particularV0

whose background valueV05X0 is time dependent. Still,the fact that the target metricFAB(V) has a nonmaximarank reflects in the appearance of an additional symmetrthe theory. It is not difficult to check that the field transfomations

Va85Va8~Vb!, Va85Va,

3-6

tio-

th

e

on

zare

-denoItnt

hiv

ee

aduace

uin

lacef

4tro

oto

latioic

ood

andardema-e

dat itsnalns

ur

umeheac-

ular

thee

so-finn

nica.

t thegi-be

dif-e

ce

ing

edrm


where a50, . . . ,3 anda51, . . . ,d, leave the action~5.1!invariant. Notice, however, that the gaugeVa50 is not al-lowed owing to the required regularity ofV ,M

A . If we restrictourselves to the small perturbations of the classical solu@VA(X)5XA1vA(X)#, we shall find that the transformations

dva5«a~x!1O~2!, dva50

leave the linearized field equations invariant. Again,fields va cannot be gauged away owing to the fact thatva

5va(x,y) while the parameters«a(x) are onlyx dependent.However, when averaged over the internal manifold, thefective fields^va& will depend on thex coordinate alone.Being a pure gauge, they will not appear in the dimensially reduced linearized equations of motion.

VI. CONCLUDING REMARKS

We have applied the ideas of geometrics models@2# tosolve the classical cosmological constant problem of KaluKlein theories. This kind of approach is not new in literatuThe authors of Refs.@3# and @4# have demonstrated howscalar fields in the form of as model can trigger spontaneous compactification. It turned out, however, that their mofailed to give massless gauge fields after the dimensioreduction. We have rewritten this theory in terms of a gemetric s model, thereby bringing it to a suggestive form.was not difficult then to realize which kind of modificatiowould reconcile the masslessness of the gauge fields withzero value of the cosmological constant. In Sec. III, tmodified theory has been proven to contain no massgauge fields. The effective 4-dimensional theory has bobtained by averaging the linearized (41d)-dimensionalequations of motion over the internal coordinates.

In search for the simple Lagrangian of the theory, we hto abandon the model of Sec. III, and look for another mofication. We have found an action functional whose eqtions of motion differ from those of Sec. III by the presen

of a term proportional to (GMN2G° MN), but retain all theirgood features. The linearized effective 4-dimensional eqtions of motion turned out to contain the standard EinsteYang-Mills and Klein-Gordon sectors. In addition, the scaexcitations of the internal manifold, in particular their tramode, have been shown to have masses of the order oPlanck mass.

In Sec. V, we have covariantized our theory. A set of1d scalar fields has been introduced in a purely geomemanner. The generally covariant theory turned out to bethe form of a nonstandards model with nonpolynomial de-pendence on the scalar field derivatives. We have demstrated how the introduction of auxiliary fields puts them ina polynomial form. Compared to geometrics models of Ref.@2#, and, in that respect, to the model of@3# and @4#, ourtheory exhibits some conceptual differences. In particuthe number of scalar fields needed for the covariantizadoes not match the rank of the background value of the Rtensor.

02500

n

e

f-

-

-.

lal-

heeen

di--

a-,

r

the

icf

n-

r,nci

The analysis done in this paper has shown all the gfeatures of the proposed Kaluza-Klein geometrics model:vanishing cosmological constant, massless gauge fields,the effective 4-dimensional theory consisting of the standEinstein, Yang-Mills and Klein-Gordon sectors. Still, all thconsiderations have been restricted to the linear approxition of the theory. What is left for future investigation are thimplications of the full interacting theory.

The classical stability of our Kaluza-Klein backgrounmetric is also to be addressed. We have already seen thlinear excitations satisfy the standard effective 4-dimensioEinstein, Yang-Mills and Klein-Gordon equations. It remaito be shown that the mass matrix in Eq.~4.4c! is positivedefinite, thereby insuring the classical linear stability of obackground metric. The answer depends on the choice ofBd.For instance, in our simple exampleBd5S2, the mass matrixturned out not to be positive definite. The classical vacustability of the full (41d)-dimensional theory is much mordifficult to achieve. It depends on both, the choice of tbackground metric, and the dynamics itself. Our simpletion functional as given by Eq.~4.1! restricts the choice ofstable internal background geometries. Whether a particinternal spaceBd is a stable solution of Eq.~4.1! should beseparately investigated. Vice versa, an arbitrary choice ofbackground spaceBd would severely restrict the form of thdynamics to stabilize it. One expects that whateverBd ischosen, there exists a~complicated enough! dynamics whichadmits it as a stable classical solution.

In search for a realistic theory of the kind, we could alfurther develop the idea of@2# to give fermions a pure geometric origin. In particular, it would be more in the spirit ogeometrics models if we chose our background metricthe form of a localized, particle-like field configuratiowhich only asymptotically approachesM43Bd. The corre-sponding theory might turn out to accommodate fermiomatter without spoiling the beauty of the Kaluza-Klein ide

The complete analysis in this paper has been done aclassical level. All the results, such as vanishing cosmolocal constant, or classical stability of the ground state, canaffected by the quantum corrections. However, being aficult task by itself, the quantization of gravity has not baddressed in this work.

ACKNOWLEDGMENTS

This work was supported in part by the Serbian ScienFoundation, Yugoslavia.

APPENDIX: AVERAGING THE FIELD EQUATIONS

In this appendix we shall give the details of the averag

procedure for the field equationsRMN5R° MN .First, we rewrite these equations in terms of the mix

components of the Ricci tensor, and bring them into the fo~2.9a!–~2.9c!. Then, we use the decompositions~2.7! and~2.8! to obtain

3-7

esp to a

paringre


R nm1

1

2w l ,n

l ,m21

2~Bmn

,m 1Bm,nm 2hn,m

m ! ;m5O~2!, ~A1a!

Fn,nmn 1R° nlB

lm1~hnn,m2hn

m,n! ,n1~w l ;nl 2wn; l

l ! ,m1Bnm; l

l2B; lnlm 5O~2!, ~A1b!

hwnm1~R° ln

mk1fmkR° nl!wkl 1w l

l;m

n2w ln

;ml2wm

l; l

n1wmn

; ll2Bn,m

m;m2B,m;nmm 1hm

m;m

n5O~2!. ~A1c!

Here,Rmn is the Ricci tensor of the 4-metricgmn , h is the corresponding d’Alembertian, andFmnl [Bm,n

l 2Bn,ml 1O(2). The

semicolon denotes covariant derivatives with respect to the internal coordinates.When projected onto the Killing basisKa

m(y), and using the decomposition~2.11! for the field variables, Eqs.~A1a!–~A1c!take the form

Rmn11

2~Ka

l Kbl!w ,mnab 1div5O~2!, ~A2a!

~Kal Kbl!F ,n

bmn12~Kal Kb

mR° lm!Abm1div5O~2!, ~A2b!

~Kal Kb

mKclKdm!hwcd1@~2Kal KclKb

mKdnR° mn1a↔b!2Kc

l KdlKamKb

nR° mn1~2Kcl Kd

nKal;mKbn;m1a↔b!2Kc

l KdlKam;nKbm;n#wcd

1~KamKb

n! ;nmhmm1~Ka

mKbnKcn1a↔b! ;mA,m

cm1div5O~2!, ~A2c!

where div denotes terms of the typeY;mm , which vanish when averaged, andFmn

a [Am,na 2An,m

a 1O(2). To get theabove result,we have used the known properties of the Killing vectors:

Kam;n1Kan;m50, Kal;mn52R° nmlk Kak .

The effective 4-equations~2.12a!–~2.12c! are obtained by averaging Eq.~A2a!–~A2c! under the assumption that our variablgmn , Am

a andwab vary slowly in y direction. Then, the average of a product is replaced by the product of averages, usmall correctionD. The coefficients in Eqs.~2.12a!–~2.12c! are given as follows:

gab[^KamKbm&, mab[^Ka

mKbnR° mn&,

sabcd[^KamKb

nKcmKdn&1a↔b,

mabcd[^4Kal K (clKb

mKd)n R° mn24K (c

l Kd)mKa

i Kbj R° i l jm1Kc

l Kdl~KanKb

m! ;mn

24K (cl Kd)n~Ka

nKbm! ;ml&1a↔b. ~A3!

One can average Eqs.~3.2a!–~3.2c! by using the preceding results, and just add the minor corrections as seen by comEqs.~3.2a!–~3.2c! with Eqs.~2.9a!–~2.9c!. The same holds for our final equations~4.2!. The corresponding mass matrices agiven by

mabcd8 [mabcd24~^Kal K (clKb

mKd)n R° mn&1a↔b!, ~A4!

mabcd9 [mabcd24lsabcd18l

d12^Ka

mKcnKbmKdn&. ~A5!

@1# T. Kaluza, Sitzungsber. K. Preuss. Akad. Wiss.K1, 966~1921!; O. Klein, Z. Phys.37, 895 ~1926!; T. Appelquist, A.Chodos and P.G. O. Freund,Modern Kaluza-Klein Theories~Addison-Wesley, Reading, MA, 1987!.

@2# M. Vasilic, Class. Quantum Grav.15, 29 ~1998!.

02500

@3# C. Omero and R. Percacci, Nucl. Phys.B165, 351 ~1980!.@4# M. Gell-Mann and B. Zwiebach, Phys. Lett.141B, 333~1984!.@5# A. Salam and J. Strathdee, Ann. Phys.~N.Y.! 141, 316~1982!.@6# M. Vasilic, Nuovo Cimento B109, 1083~1994!.

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Higher-dimensional geometric sigma models